Complete Pre-University Mathematics Curriculum

In a different thread I said

I estimate that the entirety of the average North American K-12 mathematical education can be compressed into a single book about 200 pages long, while being entirely comprehensible to a 10-year-old.

Here I will attempt to devise a concise and complete curriculum for K-12 or pre-university mathematics. I will add new material to this thread periodically, anything from lessons to exercises to bits and pieces of thoughts. It certainly won't be anything ready for publication. This thread is only meant to be a proof of concept, or a proof that a concept will fail.

Without further ado, let's get to it.


Rough outline of topics. The names of the categories and the order of topics are rather arbitrary.

Elementary topics
- Natural (positive) numbers
- Addition
- Multiplication
- Geometry -- rectangles
- Non-positive numbers
- Subtraction
- Exponents
- Base systems (10, and perhaps 2?)
- Modulo operation (despite the scary name, this simply means finding remainders)

Intermediate topics
- Parentheses
- Prime numbers, the fundamental theorem of arithmetic
- Factorization
- Division of integers, fractions, rational numbers
- Operations involving rational numbers
- Radix points (usually only base 10)
- Geometry -- triangles, parallelograms, circles (what about compass-and-straightedge constructions?)
- Single-variable algebra
- Distributive property (perhaps introduced earlier, through diagrams)
- Graphing (plotting) equations of one variable

More intermediate topics
- Combinatorics
- Set theory
- Probability theory
- Statistics
- Logarithms
- Trigonometry
- Geometry -- basic identities and proofs
- Polynomials of second degree
- Proofs by induction, and contradiction
- Systems of linear equations

Advanced topics
- Linear algebra -- vectors and matrices
- Polynomials of higher degree
- Parametric equations
- Polar coordinates
- Conics, analytic 2D geometry
- 3D geometry
- Complex numbers (and de Moivre's theorem, anyone?)

Whew, that's rather a challenge. Best tackle that one piece at a time. For example, logarithms can be summarized in a few pages. Note: the estimate of "200 pages" does not include exercises.

One way to motivate a student is to appeal to the inherent human characteristic of frugality. What number multiplied by itself equals 625? A naive but sure-fire approach would be to try 1, 2, 3 ... all the way to 25. But there clearly exist ways which require less effort.

Formal logic and proof should be taught in parallel with elementary topics.

Nothing here is final, of course.


Feel free to contribute. I know I'll be back.

I would argue that making the more advanced topics that one typically learns in Algebra II or Trigonometry class comprehensible to a 10 year old is a bit over-reaching. A 10 year old doesn't normally have the ability to deal in complex or non-concrete terms like some of those topics would be; it's unsound to expect many of them to be able to do this in terms of mental development. However, I would say that up the age to about 14 or so, and you've got a good point.

Check this sight out, it has great reference to almost any mathmatic problem you may come across at the pre-university level. Most of the program is free as well.

The Khan Academy

reply to post by socialist


Awesome idea!

As a parent of kids in public high school, I can attest the course work is meager at best. My oldest is in a level of math I learned in 7th-8th grade not so long ago. He will be a Senior next year. I makes one weep.

reply to post by socialist


AWESOME idea! I recommend lots of stars, flags, and applauses for this thread, and a sticky once it is complete!

I think you can certainly get it done. Of course, you won't be able to put in all the fillers, practice problems, short stories, and half-filled pages like the textbook makers do.

I think 200 pages is probably more than you need. I would guess K-12 mathematics would fit in probably 75 pages. What is a typical high school graduate really expected to know? It isn't very much. My brother teaches 10th grade math, and half his kids can't even multiply and divide in their heads. They have to have scratch paper for the most basic problems.

Algebra II and Trigonometry are not required to graduate in most states. They don't teach much geometry past the kindergarten level.

Add, Subtract, Multiply, DIvide, FOIL, Factor, memorize quadratic eqn but not understand it. Know about 10 basic shapes, know how to find area of a rectangle and triangle, know how to find circumference and area of a circle. That right there is probably as much as most high school graduates retain.

I'll guarantee right now, that by the end of 1st grade my oldest son can pass the math portion of the GED. I'll bet by the end of 5th grade he will be able to pass the whole thing.

To the OP. Don't you want to adopt my 18 year old son?
He'll have to do math and science next year as a bridging coarse for university!
I gave up on math when they tried to teach me algabra! Never could get why you -+x or devide x's and y's ec. Lol

Although your idea is brilliant, it simply wouldn't fly. Especially in California. Here is a detailed review of a recent California math textbook.

Addison-Wesley Secondary Math: An Integrated Approach: Focus on Algebra

WARNING! It is not for the faint of heart. Personally, it makes me feel sick to my stomach.

Oh, and yes, the review is a bit dated - but take into account that the textbooks in classrooms are, too.

Apologies all around -- long story short, that was an absence I would rather have not happened. My biggest regret is not writing anything down. But now that I'm back I should be able to contribute again. If I realize it's better to do so, then I'll revive this in a new thread.

I would argue that making the more advanced topics that one typically learns in Algebra II or Trigonometry class comprehensible to a 10 year old is a bit over-reaching. A 10 year old doesn't normally have the ability to deal in complex or non-concrete terms like some of those topics would be; it's unsound to expect many of them to be able to do this in terms of mental development. However, I would say that up the age to about 14 or so, and you've got a good point.

Point taken. It may be true that a younger person simply doesn't have the experience to appreciate math as much as an older person. But I don't think there is an arbitrary age when, say, it's suddenly possible for me to "fully understand trig".

After some more consideration however, I figured out that my intention is not to force math down people's throats. Perhaps this is a quest which is doomed to fail. (Try nonetheless.)

Counting is something taught very on, but it might be worth at least revisiting it, taking a critical look at it. How do we count the way we count? Clearly we do it by virtue of our ten-fingered hands. But it's pretty senseless to teach people that if Martians had eight fingers on each hand they'd be able to somehow count in hex -- they're not Martians and nor will they ever be. That's silly. Instead we should give them some motivation.

Early humans may have used small symbols such as stones to represent livestock, for purposes of trading. This is because it's easier to handle small symbols than to handle livestock. This, for example, gives an immediate motivation for using abstract things to represent real things -- you can manipulate them much more easily.

Now things get trickier. Let's say one symbol represents one cow. Then you'd need 100 symbols to represent 100 cows. Counting 100 symbols is just as difficult as counting 100 cows, so that defeats the whole purpose. A clever solution could be to use a different kind of symbol, perhaps a larger stone, to represent 20 cows. Then you'd need only 5 of those symbols to represent those 100 cows. Plus it offers you the flexibility to combine the 2 kinds of symbols; for example, 1 large and 5 small to represent 25 cows.

For most practical purposes this works fine. But then civilization expands, and we have to deal with much larger quantities. Let's say you had to count the people in a town. There could be over a thousand people in a town, which means the same problem occurs: counting becomes difficult. No problem, just invent more symbols. Still, we can't continue inventing symbols like this.

That is when the decimal system comes in. It's such a simple thing in concept: a set of rules for creating new symbols. (Almost analogous to a programming language.) We specify a base, such as 10. We use one kind of symbol to represent numbers up to that base: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Then we run out, and this fulfills a condition to create a new symbol, or place value. In a sense the tens place is a different kind of symbol from the ones place. We continue in this fashion: 10, 11, 12, ..., 97, 98, 99. We run out again, and so we create another new symbol, the hundreds place.

One might notice that certain elegant patterns emerge when manipulating this system, and that the need to create a new symbol decreases as numbers get larger and larger.

It's possible that the above lesson would be lost in some way, can not be fully appreciated without a deep understanding of addition, multiplication, exponentiation. There can be other problems. It might not even be useful to teach from this angle -- one can be an effective engineer, for example, without ever knowing why the base system exists. It might belong elsewhere that is not a math curriculum.

Thoughts on this? I'm interested.

This sounds like a very interesting proposition. I don't understand all of your phrases, but have been pretty much a math whiz all of my life. I always gave credit to genetics, my dad was a math whiz.

I think, as a mathematics tutor, that the real need is to introduce people to concepts earlier than is done now. A large part of my success with students I attribute to the fact that I don't try to 'hide' higher-level concepts... I just don't try to force them before the student is ready for them either.

Example: You start using base systems when you learn to carry and borrow. Point out why you carry the one instead of a ten, and why when you borrow one it becomes a ten. Then drop it. That little bit of information will stew in the mind for a while and by the time they are ready to study bases, they already have the foundation needed to understand the concepts.

Example: Subtraction is simply adding a negative. It should be presented as such, not after subtraction is learned, but when real numbers are learned, which is very early on. Division is multiplication by the inverse, and should be taught as such.

I honestly believe if these and similar changes were made to our educational system, all kids could easily graduate the 12th grade adept at differentiation and integration at the least... not trying to struggle "how do you add letters and get a number?"

TheRedneck

reply to post by TheRedneck


If I'm thinking correctly, you present concepts to students when they are needed. But what about the "needed" part? Why do students need it? If you are a math tutor then your students need the concepts to satisfy some requirement put forth by the school board. Actually, I suppose that's fair. Taking this approach, one can start listing the minimum requirements to graduate from high school and think up lessons from there.

Perhaps this is a product of math education, but I find that people do not think commutatively. Subtraction and division may just be addition and multiplication in disguise, but people find them harder than addition and multiplication. Simple thought experiment:

10 - 2 = ?
10 + (-2) = ?

Equivalent, but the second expression is harder to evaluate. Why? There are more symbols. Another thought experiment:

What is 50% of 18?
What is 18% of 50?

Again, equivalent, but the second expression is harder to evaluate. Actually, try this with any value of X instead of 18.

Somewhat of an actual lesson plan, hooray!

Exponents (with the assumption that previous knowledge has been attained).

Addition is shorthand for counting by ones. Multiplication is shorthand for addition. Is there a shorthand for multiplication? What if you wanted to say "2 * 2 * 2 * 2" quickly and concisely? You can say "2 times itself 4 times." We say "2 to the power of 4." Since we are only concerned with the thing we wish to multiply and the number of times we wish to multiply it, we can write it more concisely as

a^b

where a is the thing we wish to multiply and b is the number of times. We can now say that (examples follow)

2^4 = 16
3^3 = 27
10^2 = 100

(Good time for some good old-fashioned computation drills?)

An important identity is a^b * a^c = a^(b + c). a^b / a^c = a^(b - c) should naturally follow. Then a^0 = 1 (and an explanation of why).

It's reasonable to expect that 0^b = 0 for any b, since 0 multiplied by itself is always 0. But what about 0^0? (Here be dragons.)

Multiplication also applies to the exponents: (a^b)^c = a^(b * c). But what about the pitfall of a^(b^c)?

Fractional and negative exponents are also possible. Explore what one can do with it...

Golly, this will turn into quite a lot of material, won't it? I would be hard-pressed to present all this in a concise, palatable manner. Or perhaps I shouldn't worry too much about the abilities of students -- will they cope given whatever tools they have? Sure as hell dumbing things down doesn't help students one bit.

I've now got a wiki hosted on Wikia where people can brainstorm more easily and with LaTeX support: link. Feel free to use the wiki.

For some reason exponents just strikes me as an "easy" place to start...

reply to post by socialist


I love what you are doing here!

I think that if more take action like this, including more advanced subjects, then we will all benefit. Learning and education is so fundamental to a society, but it seems to be quite overlooked, and a lot of what is "learned" is either biased or outdated by the time they finish High School or College.

The things that dont change are patterns (math), and the approach taught in the core scientific method.

The further we go in attempting to make this available to anyone and everyone is a great thing. If we were to even just start with these basics in public schools, the overall learning ability would increase drastically. Teach them how to learn and how to best express it, rather than what to learn to score high on some arbitrary test.

Anywho, thank you for putting something like this out there. Open source knowledge and teaching could be fundamental to a stronger nation, and even world.

reply to post by socialist

I think about math as something more important than simply a need to satisfy a school requirement. Math is the language of nature, the language of the Universe, and, if I can be so bold, therefore the language of God. Everything you do in life can be helped with mathematical fluency.

10 - 2 = ?
10 + (-2) = ?

You see, I see both problems as the same. The second poses no more problem to me than the first, because I immediately mentally condense the two signs into one. All terms are added values; the only reason a '+' sign is used on positive terms is to separate them.

Addition does not have to be specified, in other words. It is assumed between terms and the separating signs are nothing more than separators. Only the negative (subtraction) is actually meaningful.

What is 50% of 18?
What is 18% of 50?

Yes, both identical, and understanding the concept of multiplication/division makes that clear almost immediately. The first is 50·18/100, the second is 18·50/100... both the same under the communicative law.

That's a good explanation of what I mean... learn the language and reading becomes simple... memorize a bunch of answers and it is painfully difficult. We need much more emphasis on how to speak math and much less emphasis on memorizing how to work a problem.

TheRedneck

I see that you started this thread back in 2012? Are you still working on this by any chance?

I am actually quite interested and would like to know how this turns out if you are.

reply to post by LegendaryPotato


I was MIA in 2012. I am working on it now whenever I'm not terribly busy.

A wiki has been set up, so anyone who is interested can contribute.

Concise Math Wiki

You will need more than 200 pages, but it's a good idea to put it all in one book (or PDF).